The Gaussian-like Normalization Constant
Jason D. M. Rennie
jrennie@gmail.com
November 6, 2005
Abstract
Here we give the procedure for deriving the normalization constant for
a Gaussian-like distribution
Consider the function described by
P̃ (x) = exp −x2 .
(1)
Clearly this can be normalized to a distribution since P̃ (x) < exp(−x) for |x| ≥ 1
and P̃ (x) ≤ 1 for |x| ≤ 1. In fact, P̃ (x) is (nearly) the uni-variate un-normalized
standard Normal distribution (Gaussian with zero mean and unit variance).
The standard normal is a very well known distribution, but calculation of
the normalization constant isn’t completely trivial. The normalization constant
is P̃ integrated from −∞ to +∞. But, no antiderivative exists for exp(−x2 ).
The trick is to consider the bi-variate version of P̃ and to integrate using
polar coordinates. Define
Q̃(x, y) = P̃ (x)P̃ (y) = exp(−x2 ) exp(−y 2 ) = exp(−(x2 + y 2 )).
(2)
RR
To normalize Q̃, we must calcualte ZQ =
exp(−x2 − y 2 )dxdy, where both
integrals range over R. As is well known, any bi-variate integral over Euclidean
coordinates can be rewritten using polar coordinates (e.g. §17.3 of [1]),
ZZ
ZZ
f (x, y)dydx =
f (r cos θ, r sin θ)rdrdθ.
(3)
In our case, f (r cos θ, r sin θ) = exp(−r2 ), so the normalization constant can be
written as
Z 2πZ ∞
ZQ =
exp(−r2 )rdrdθ.
(4)
0
0
Since
R ∞the inner integral is constant as a function of θ, we can rewrite it as ZQ =
2π 0 exp(−r2 )rdr. And, unlike the original integral we posed, this integral
is trivial and has an immediately apparante anti-derivative. d exp(−r2 ) =
−2r exp(−r2 )dr, so
ZQ = −π e−∞ − e−0 = π.
(5)
1
We’re almost done. Define Q(x, y) = Q̃(x,y)
= √1π exp(−x2 ) √1π exp(−y 2 ),
ZQ
which is a distribution.
Due to symmetry, it must be that the normalization
√
constant for P̃ is π. Thus, P (x) = √1π exp(−x2 ) is a distribution.
It is straightforward to extend this result to multiple dimensions. Let ~x ∈ Rn .
Then
R(~x) = π −n/2 exp(−k~xk22 ),
P
is a distribution, where k~xk22 = i x2i is the squared L2 -norm.
Thanks to John for reminding me of the polar coordinates trick.
References
[1] E. W. Swokowski. Calculus. PWS-Kent Publishing Company, 1983.
2
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